On universal inference in Gaussian mixture models

A recent line of work provides new statistical tools based on game-theory and achieves safe anytime-valid inference without assuming regularity conditions. In particular, the framework of universal inference proposed by Wasserman, Ramdas and Balakrishnan [78] offers new solutions to testing problems by modifying the likelihood ratio test in a data-splitting scheme. In this paper, we study the performance of the resulting split likelihood ratio test under Gaussian mixture models, which are canonical examples for models in which classical regularity conditions fail to hold. We establish that under the null hypothesis, the split likelihood ratio statistic is asymptotically normal with increasing mean and variance. Contradicting the usual belief that the flexibility of universal inference comes at the price of a significant loss of power, we prove that universal inference surprisingly achieves the same detection rate $(n^{-1}\log\log n)^{1/2}$ as the classical likelihood ratio test.